# NAME

Math::PlanePath::KochPeaks -- Koch curve peaks

# SYNOPSIS

`````` use Math::PlanePath::KochPeaks;
my \$path = Math::PlanePath::KochPeaks->new;
my (\$x, \$y) = \$path->n_to_xy (123);``````

# DESCRIPTION

This path traces out concentric peaks made from integer versions of the self-similar `KochCurve` at successively greater replication levels.

``````                               29                                 9
/  \
27----28    30----31                        8
\              /
23          26          32          35               7
/  \        /              \        /  \
21----22    24----25                33----34    36----37      6
\                                                  /
20                                              38         5
/                                                  \
19----18                                        40----39      4
\                                      /
17                 8                41               3
/                 /  \                 \
15----16           6---- 7     9----10          42----43      2
\                 \              /                 /
14                 5     2    11                44         1
/                 /     /  \     \                 \
13                 4     1    3     12                45  <- Y=0

^
-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9 ...``````

The initial figure is the peak N=1,2,3 then for the next level each straight side expands to 3x longer with a notch in the middle like N=4 through N=8,

``````                                  *
/ \
*---*     becomes     *---*   *---*``````

The angle is maintained in each replacement so

``````                                  *
/
*---*
\
*                     *
/        becomes      /
*                     *``````

For example the segment N=1 to N=2 becomes N=4 to N=8, or in the next level N=5 to N=6 becomes N=17 to N=21.

The X,Y coordinates are arranged as integers on a square grid. The result is flattened triangular segments with diagonals at a 45 degree angle.

Unlike other triangular grid paths `KochPeaks` uses the "odd" squares, with one of X,Y odd and the other even. This means the rotation formulas etc described in "Triangular Lattice" in Math::PlanePath don't apply directly.

## Level Ranges

Counting the innermost N=1 to N=3 peak as level 0, each peak is

``````    Nstart = level + (2*4^level + 1)/3
Nend   = level + (8*4^level + 1)/3
points = Nend-Nstart+1 = 2*4^level + 1``````

For example the outer peak shown above is level 2 starting at Nstart=2+(2*4^2+1)/3=13 through to Nend=2+(8*4^2+1)/3=45 with points=2*4^2+1=33 inclusive (45-13+1=33). The X width at a given level is the endpoints at

``````    Xlo = -(3^level)
Xhi = +(3^level)``````

For example the level 2 above runs from Xlo=-9 to Xhi=+9. The highest Y is the centre peak half-way through the level at

``````    Ypeak = 3^level
Npeak = level + (5*4^level + 1)/3``````

For example the level 2 outer peak above is Ypeak=3^2=9 at N=2+(5*4^2+1)/3=29. For each level the Xlo,Xhi and Ypeak extents grow by a factor of 3.

The triangular notches in each segment are not big enough to go past the Xlo and Xhi end points. The new triangular part can equal the ends, such as N=6 or N=19, but not go beyond.

In general a segment like N=5 to N=6 which is at the Xlo end will expand to give two such segments and two points at the limit in the next level, as for example N=5 to N=6 expands to N=19,20 and N=20,21. So the count of points at Xlo doubles each time,

``````    CountLo = 2^level
CountHi = 2^level      same at Xhi``````

# FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

`\$path = Math::PlanePath::KochPeaks->new ()`

Create and return a new path object.

`(\$x,\$y) = \$path->n_to_xy (\$n)`

Return the X,Y coordinates of point number `\$n` on the path. Points begin at 0 and if `\$n < 0` then the return is an empty list.

Fractional `\$n` gives an X,Y position along a straight line between the integer positions.

## Level Methods

`(\$n_lo, \$n_hi) = \$path->level_to_n_range(\$level)`

Return per "Level Ranges" above,

``````    ((2 * 4**\$level + 1)/3 + \$level,
(8 * 4**\$level + 1)/3 + \$level)``````

# FORMULAS

## Rectangle to N Range

The baseline for a given level is along a diagonal X+Y=3^level or -X+Y=3^level. The containing level can thus be found as

``````    level = floor(log3( Xmax + Ymax ))
with Xmax as maximum absolute value, max(abs(X))``````

The endpoint in a level is simply 1 before the start of the next, so

``````     Nlast = Nstart(level+1) - 1
= (level+1) + (2*4^(level+1) + 1)/3 - 1
= level + (8*4^level + 1)/3``````

Using this Nlast is an over-estimate of the N range needed, but an easy calculation.

It's not too difficult to work down for an exact range, by considering which parts of the curve might intersect a rectangle. But some backtracking and level descending is necessary because a rectangle might extend into the empty part of a notch and so be past its baseline but not intersect any. There's plenty of room for a rectangle to intersect nothing at all too.